A New Method for Laplace Transforms of Multi-Term Fractional Differential Equations of the Caputo Type

The Laplace transform method is one of the powerful tools in studying the frac- tional differential equations (FDEs). In this paper, it is shown that the Heaviside expansion method for integer order differential equations is also applicable to the Laplace transforms of multi-term Caputo fractional differential equations (FDEs) of zero initial conditions if the orders of Caputo derivatives are integer multiples of a common real number. The particular solution of a linear multi-term Caputo FDE is obtained by its Laplace transform and the Heaviside expansion method. A Caputo FDE of non zero initial conditions is transformed to an Caputo FDE of zero initial conditions by an appropriate change of variables. In the latter, the terms originated from the initial conditions appear as nonhomogeneous terms. Thus, the Caputo FDE of nonzero initial conditions is obtained as the particular solutions to the equivalent Caputo FDE of zero initial conditions. The solutions of a linear multi-term Caputo FDEs of nonzero initial conditions are expressed through the two parameter Mittag-Leffler functions.